Category Archives: Networks

When we talk about strategic equilibrium, we can talk in a very formal sense, as many refinements with their well-known epistemic conditions have been proposed, the nature of uncertainty in such equilibria has been completely described, the problems of sequential decisionmaking are properly handled, etc. So when we do analyze history, we have a useful tool to describe how changes in parameters altered the equilibrium incentives of various agents. Path dependence, the idea that past realizations of history matter (perhaps through small events, as in Brian Arthur’s work) is widespread. A typical explanation given is increasing returns. If I buy a car in 1900, I make you more likely to buy a car in 1901 by, at the margin, lowering the production cost due to increasing returns to scale or lowering the operating cost by increasing incentives for gas station operators to operate.

This is quite informal, though; worse, the explanation of increasing returns is neither necessary nor sufficient for history-dependence. How can this be? First, consider that “history-dependence” may mean (at least) six different things. History can effect either the path of history, or its long-run outcome. For example, any historical process satisfying the assumptions of the ergodic theorem can be history-dependent along a path, yet still converge to the same state (in the network diffusion paper discussed here last week, a simple property of the network structure tells me whether an epidemic will diffuse entirely in the long-run, but the exact path of that eventual diffusion clearly depends on something much more complicated). We may believe, for instance, that the early pattern of railroads affected the path of settlement of the West without believing that this pattern had much consequence for the 2010 distribution of population in California. Next, history-dependence in the long-run or short-run can depend either on a state variable (from a pre-defined set of states), the ordered set of past realizations, or the unordered set of past realizations (the latter called path and phat dependence, respectively, since phat dependence does not depend on order). History matters in elections due to incumbent bias, but that history-dependence can basically be summed up by a single variable denoting who is the current incumbent, omitting the rest of history’s outcomes. Phat dependence is likely in simple technology diffusion: I adopt a technology as a function of which of my contacts has adopted it, regardless of the order in which they adopted. Path dependence comes up, for example, in models of learning following Aumann and Geanakoplos/Polemarchakis, consensus among a group can be broken if agents do not observe the time at which messages were sent between third parties.

Now consider increasing returns. For which types of increasing returns is this necessary or sufficient? It turns out the answer is, for none of them! Take again the car example, but assume there are three types of cars in 1900, steam, electric and gasoline. For the same reasons that gas-powered cars had increasing returns, steam and electric cars do as well. But the relative strength of the network effect for gas-powered cars is stronger. Page thinks of this as a biased Polya process. I begin with five balls, 3 G, 1 S and 1 E, in an urn. I draw one at random. If I get an S or an E, I return it to the urn with another ball of the same type (thus making future draws of that type more common, hence increasing returns). If I draw a G, I return it to the urn along with 2t more G balls, where t is the time which increments by 1 after each draw. This process converges to having arbitrarily close to all balls of type G, even though S and E balls also exhibit increasing returns.

Why about the necessary condition? Surely, increasing returns are necessary for any type of history-dependence? Well, not really. All I need is some reason for past events to increase the likelihood of future actions of some type, in any convoluted way I choose. One simple mechanism is complementarities. If A and B are complements (adopting A makes B more valuable, and vice versa), while C and D are also complements, then we can have the following situation. An early adoption of A makes B more valuable, increasing the probability of adopting B the next period which itself makes future A more valuable, increasing the probability of adopting A the following period, and so on. Such reasoning is often implicit in the rhetoric linking market-based middle class to a democratic political process: some event causes a private sector to emerge, which increases pressure for democratic politics, which increases protection of capitalist firms, and so on. As another example, consider the famous QWERTY keyboard, the best-known example of path dependence we have. Increasing returns – that is, the fact that owning a QWERTY keyboard makes this keyboard more valuable for both myself and others due to standardization – is not sufficient for killing the Dvorak or other keyboards. This is simple to see: the fact that QWERTY has increasing returns doesn’t mean that the diffusion of something like DVD players is history-dependent. Rather, it is the combination of increasing returns for QWERTY and a negative externality on Dvorak that leads to history-dependence for Dvorak. If preferences among QWERTY and Dvorak are Leontief, and valuations for both have increasing returns, then I merely buy the keyboard I value highest – this means that purchases of QWERTY by others lead to QWERTY lock-in by lowering the demand curve for Dvorak, not merely by raising the demand curve for QWERTY. (And yes, if you are like me and were once told to never refer to effects mediated by the market as “externalities”, you should quibble with the vocabulary here, but the point remains the same.)

All in all interesting, and sufficient evidence that we need a better formal theory and taxonomy of history dependence than we are using now.

Final version in the QJPS (No IDEAS version). The essay is written in a very qualitative/verbal manner, but more because of the audience than the author. Page graduated here at MEDS, initially teaching at Caltech, and his CV lists quite an all-star cast of theorist advisers: Myerson, Matt Jackson, Satterthwaite and Stanley Reiter!

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No need to separate economics from the rest of the social sciences, and no need to separate social science from the rest of science: we often learn quite a bit from our compatriot fields. Here’s a great example. Consider any epidemic diffusion, where a population (of nodes) is connected to each other (along, in this case, unweighted edges, equal to 1 if and only if there is a link between the nodes). Consider the case where nodes can become “infected” – in economics, we may think of nodes as people or cities adopting a new technology, or purchasing a new product. Does a given seeding on the network lead to an “infection” that spreads across the network, or is the network fairly impervious to infections?

This seems like it must be a tricky question, for nodes can be connected to other nodes in an arbitrary fashion. Let’s make it even more challenging for the analyst: allow there to be m “susceptible” states, an “exposed” state, an “infected” state, and N “vaccinated” states, who cannot be infected. Only exposed or infected agents can propagate an infection, and do so to each of their neighbors in any given period according to probabilities a and b, independently across neighbors. Parameters tell me the probability each agent transitions from susceptible or vaccinated states to other such states.

You may know the simple SIR model – susceptible, infected, recovered. In these models, all agents begin as susceptible pr infected. If my neighbor is infected and I am susceptible, he gives me the disease with probability a. If I am infected, I recover with probability c. This system spreads across the population if the first eigenvalue of the adjacency matrix (which equals 1 if two people are connected, and 0 otherwise) is greater than a/c. (Incredibly, I believe this proof dates back to Kermack and McKendrick in 1927). That is, the only way the network topology matters is in a single-valued summary statistic, the first eigenvalue. Pretty incredible.

The authors of the present paper show that this is a general property. For any epidemic model in which disease spreads over a network such that, first, transmissions are independent across neighbors, and second, one can only enter the exposed or infected state from an exposed or infected neighbor, the general property is the same: the disease spreads through the population if the first eigenvalue of the adjacency matrix is larger than a constant which depends only on model parameters and not on the topology of the network (and, in fact, these parameters are easy to characterize). It is a particularly nice proof. First we compute the probabilities of transitioning from each state to any other. This gives us a discrete-time nonlinear dynamic system. Such systems are asymptotically stable if all real eigenvalues of the nonlinear dynamic are less than one in absolute value. If there are no infections at all, the steady state is just the steady state of a Markov chain: only infected or exposed people can infect me, so the graph structure doesn’t matter if we assume no infections, and transition between the susceptible and vaccinated states are just Markov by assumption. We then note that the Jacobian has a nice block structure which limits the eigenvalues to being one of two types, show that the first type of eigenvalues are always less than one in absolute value, then show that the second types are less than one if and only if a property depending on model parameters only are satisfied; this property has nothing to do with the network topology.

The result tells you some interesting things as well. For example, say you wish to stop the spread of an epidemic. Should you immunize people with many friends? No – you should immunize the person who lowers the first eigenvalue of the adjacency matrix the most. This result is independent of the actual network topology or the properties of the disease (how long it incubates, how fast it transmits, how long people stay sick, how likely they are to develop natural immunity, etc.). Likewise, in the opposite problem, if you wish an innovation to diffuse through a society, how should you organize conferences or otherwise create a network? Create links between people or locations such that the first eigenvalue of the adjacency matrix increases by the highest amount. Again, this is independent of the current network topology or the properties of the particular invention you wish to diffuse. Nice.

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Petra Persson is on the job market this year from Columbia. Her CV is pretty incredible – there’s pure theory, cutting edge empirical techniques, policy work, networks, behavioral and more. Her job market paper is about the impact of social insurance policy on seemingly unrelated markets like marriage, and I’ll discuss it briefly at the end of the post, but I want to focus on another paper on hers which struck me as quite interesting.

Imagine a benevolent ruler who has private information about some policy, such as the relative safety of wearing seatbelts. This ruler can either tell citizens the information, or lie, or coerce them to take such action. Naive libertarianism suggests that we should always be truthful is altruistic; consumers can then weigh the information according to their preferences and then choose the policy optimal for them.

But note something interesting. On some issues, one subset of politicians has libertarian leanings, while on others, a different subset has those leanings. For instance, a politician may favor legal assisted suicide but insist on mandatory seatbelt rules, while another politician may be against the mandatory belt and also against legal assisted suicide. Politicians can even vary in how libertarian they wish to be depending on who the policy affects. Witness that many politicians favor legalizing marijuana but very few favor legalizing it for 16 year olds. What explains this behavior?

Loginova and Persson examine this theoretically. Take a population of citizens. There are two possible states, 0 and 1. They can either think each state equally likely yet have different heterogeneous preferences from the politician (measured with a Crawford-Sobel style quadratic loss, though this isn’t a critical model) or they can have identical preferences as the politician yet have different heterogenous (prior) beliefs about the probability of each state. The politician can be altruistic to varying degrees – more altruism means he, according to his own prior, puts more and more weight on the utility of the agent. The politician gets a noisy signal about the true state. To limit the extent of opposing beliefs, the politician is restricted to having the same prior as the median citizen.

If the politician can only advise or not advise, when does he make a truthful public announcement? If he disagrees on preferences with the citizens, then the more altruistic, the more likely he is to announce truthfully, for the standard libertarian reason: the citizens know their own preferences, and the better informed they are, the better they can maximize their own welfare. If, however, he disagrees on priors with the citizens, then the more altruistic, the less likely he is to announce truthfully: altruism means I care about the citizen’s welfare, but since they have priors that are in my eyes wrong, the citizens know that even when I am altruistic I have incentive to lie so that citizens take actions that are optimal according to my prior, therefore truthful communication cannot be sustained.

Now what if the politician could (at a cost to him) force all individuals to take an individual action? With preference disagreement, an altruistic politician would never do this, both because he can send all the information to citizens with a free message and also because a mandate does not respect heterogeneity of preferences. Even if action 0 is better than action 1 for 90% of the population, an altruistic principle also cares about the other 10%. With disagreement about priors, however, an altruistic politician is more likely to impose a mandate the more altruistic he is. Even though citizens have heterogeneous priors, the principle thinks all of them are wrong, and hence is not worried about heterogeneity when imposing a mandate. Since we noted in the last paragraph that altruistic politicians who have different priors from citizens will not be able to credibly send their information, the mandate allows the politician’s private information to be used in the citizen’s actions.

Finally, what if the politician can send individual-level messages or enforce individual mandates? A politician with preference disagreement need to be fairly altruistic before his public message is credible; in fact, he needs to be able to credibly persuade the individual with the average disagreement in order for his public signal to be credible. If he is not altruistic enough, he can still credibly persuade those agents who have only a limited amount of preference disagreement with him. If mandates are possible, the politician with limited altruism will force individuals whose preferences are very different from the politician to take the politician’s desired action, but since preferences of the politician and the agents are more aligned when altruism is higher, the share of citizens who face a mandate declines as the politician’s altruism increases. Likewise, a politician with disagreement about priors can only truthfully send information when his altruism is low. If the politician is very altruistic, even though the public signal will not be believed, a politician can still credibly send information to those whose priors are similar to the politician. The politician with low levels of altruism will only mandate the action of agents with extreme beliefs, but as altruism increases, more and more citizens will face a mandate.

Very good – the use of paternalistic policies, and the extent to which they are targeted at individuals, depends qualitatively on whether the politician disagrees with the agents about their preferences or about their knowledge, and the extent to which mandates are applied on certain groups depends on how extreme their preferences or beliefs are. There is nothing inherently contradictory in an altruistic politician taking the libertarian side on one issue and the paternalistic side on another.

July 2012 working paper (No IDEAS version). Petra has many other interesting papers. In her job market paper, presented here last week, she shows that social insurance, in this case a widow’s benefit in Sweden, can have major affects in other markets. In particular, a really nice regression discontinuity shows that the benefit was leading to a huge number of extra marriages, that these were more likely to end in divorce, that intrahousehold bargaining was affected, and much more (Jeff at Cheap Talk has a longer description). Her paper Circles of Trust notes a reason for cliquish behavior in some labor markets. If I have information whose value declines with use (such as a stock tip) and I am altruistic, I may wish to tell my friends the info. But I worry that they will tell their friends, who I don’t know and hence don’t really care about. If my friend could commit not to tell his friends, I would give him the info. How can we commit ex-ante? Make our social networks a clique. I would bet that this phenomenon explains hiring in, say, small hedge funds to a great extent.

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Job market talks for 2012 have concluded at many schools, and therefore this is my last post on a job candidate paper. This is also the only paper I didn’t have a change to see presented live, and for good reason: Melissa Dell is clearly this year’s superstar, and I think it’s safe to assume she can have any job she wants, and at a salary she names. I have previously discussed another paper of hers – the Mining Mita paper – which would also have been a mindblowing job market paper; essentially, she gives a cleanly identified and historically important example of long-run effects of institutions a la Acemoglu and Robinson, but the effect she finds is that “bad” institutions in the colonial era led to “good” outcomes today. The mechanism by which historical institutions persist is not obvious and must be examined on a case-by-case basis.

Today’s paper is about another critical issue: the Mexican drug war. Over 40,000 people have been killed in drug-related violence in Mexico in the past half-decade, and that murder rate has been increasing over time. Nearly all of Mexico’s domestic drug production, principally pot and heroin, is destined for the US. There have been suggestions, quite controversial, that the increase in violence is a result of Mexican government policies aimed at shutting down drug gangs. Roughly, some have claimed that when a city arrests leaders of a powerful gang, the power vacuum leads to a violent contest among new gangs attempting to move into that city; in terms of the most economics-laden gang drama, removing relatively non-violent Barksdale only makes it easier for violent Marlo.

But is this true? And if so, when is it true? How ought Mexico deploy scarce drugfighting resources? Dell answers all three questions. First, she notes that the Partido Acción Nacional is, for a number of reasons, associated with greater crackdowns on drug trafficking in local areas. She then runs a regression discontinuity on municipal elections – which vary nicely over time in Mexico – where PAN barely wins versus barely loses. These samples appear balanced according to a huge range of regressors, including the probability that PAN has won elections in the area previously, a control for potential corruption at the local level favoring PAN candidates. In a given municipality-month, the probability of a drug-related homicide rises from 6 percent to 15 percent following a PAN inauguration after such a close election. There does not appear to be any effect during the lame duck period before PAN takes office, so the violence appears correlated to anti-trafficking policies that occur after PAN takes control. There is also no such increase in cases where PAN barely loses. The effect is greatest in municipalities on the border of two large drug gang territories. The effect is also greatest in municipalities where detouring around that city on the Mexican road network heading toward the US is particularly arduous.

These estimates are interesting, and do suggest that Mexican government policy is casually related to increasing drug violence, but the more intriguing question is what we should do about this. Here, the work is particularly fascinating. Dell constructs a graph where the Mexican road network forms edges and municipalities form vertices. She identifies regions which are historical sources of pot and poppyseed production, and identifies ports and border checkpoints. Two models on this graph are considered. In the first model, drug traffickers seek to reach a US port according to the shortest possible route. When PAN wins a close election, that municipality is assumed closed to drug traffic and gangs reoptimize routes. We can then identify which cities are likely to receive diverted drug traffic. Using data on drug possession arrests above $1000 – traffickers, basically – she finds that drug confiscations in the cities expected by the model to get traffic post-elections indeed rises 18 to 25 percent, depending on your measure. This is true even when the predicted new trafficking routes do not have a change in local government party: the change in drug confiscation is not simply PAN arresting more people, but actually does seem like more traffic along the route.

A second model is even nicer. She considers the equilibrium where traffickers try to avoid congestion. That is, if all gangs go to the same US port of entry, trafficking is very expensive. She estimates a cost function using pre-election trafficking data that is fairly robust to differing assumptions about the nature of the cost of congestion, and solves for the Waldrop equilibrium, a concept allowing for relatively straightforward computational solutions to congestion games on a network. The model in the pre-election period for which parameters on the costs are estimated very closely matches actual data on known drug trafficking at that time – congestion at US ports appears to be really important, whereas congestion on internal Mexican roads doesn’t matter too much. Now again, she considers the period after close PAN elections, assuming that these close PAN victories increase the cost of trafficking by some amount (results are robust to the exact amount), and resolves the congestion game from the perspective of the gangs. As in the simpler model, drug trafficking rises by 20 percent or so in municipalities that gain a drug trafficking route after the elections. Probability of drug-related homicides similarly increases. A really nice sensitivity check is performed by checking cocaine interdictions in the same city: they do not increase at all, as expected by the model, since the model maps trafficking routes from pot and poppy production sites to the US, and cocaine is only transshipped to Mexico via ports unknown to the researcher.

So we know now that, particularly when a territory is on a predicted trafficking route near the boundary of multiple gang territories, violence will likely increase after a crackdown. And we can use the network model to estimate what will happen to trafficking costs if we set checkpoints to make some roads harder to use. Now, given that the government has resources to set checkpoints on N roads, with the goal of increasing trafficking costs and decreasing violence, where ought checkpoints be set? Exact solutions turn out to be impossible – this “vital edges” problem in NP-hard and the number of edges is in the tens of thousands – but approximate algorithms can be used, and Dell shows which areas will benefit most from greater police presence. The same model, as long as data is good enough, can be applied to many other countries. Choosing trafficking routes is a problem played often enough by gangs that if you buy the 1980s arguments about how learning converges to Nash play, then you may believe (I do!) that the problem of selecting where to spend government counter-drug money is amenable to game theory using the techniques Dell describes. Great stuff. Now, between the lines, and understand this is my reading and not Dell’s claim, I get the feeling that she also thinks that the violence spillovers of interdiction are so large that the Mexican government may want to consider giving up altogether on fighting drug gangs.

http://econ-www.mit.edu/files/7484 (Nov 2011 Working Paper. I should note that this year is another example of strong female presence at the top of the economics job market. The lack of gender diversity in economics is problematic for a number of reasons, but it does appear things are getting better: Heidi Williams, Alessandra Voena, Melissa Dell, and Aislinn Bohren, among others, have done great work. The lack of socioeconomic diversity continues to be worrying, however; the field does much worse than fellow social sciences at developing researchers hailing from the developing world, or from blue-collar family backgrounds. Perhaps next year.)

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Ah…strategic action on network topologies. There is a wily problem. Tons of work has gone into the problem of strategic action on networks in the past 15 years, and I think it’s safe to say that the vast majority is either trivial or has proved too hard of a problem to say anything useful at all. This recent paper by Jeanne Hagenbach is a nice exception: it’s not all obvious, and it addresses an important question.

There is a fairly well-known experimental paper by Bonacich in the American Sociological Review from 1990 in which he examines how communications structure affects the centralizing of information. A group of N players attempt to gather N pieces of information (for example, a 10-digit string of numbers). They each start with one piece. A communication network is endowed on the group. Every period, each player can either share each piece of information they know with everyone they are connected to, or hide their information. When some person collects all the information, a prize is awarded to everybody, and the size of the prize decreases in the amount of time it took to gather the info. The person (or persons) who have all of the information in this last period are awarded a bonus, and if there are multiple solvers in the final period, the bonus is split among them. Assume throughout that the communications graph is undirected and connected.

Hagenbach formalizes this paper as a game, using SPNE instead of Nash as a solution concept in order to avoid the oft-seen problem of networks where “everybody do nothing” is an equilibrium. She proves the following. First, if the maximum game length is at least N-1 periods, then every SPNE involves information being aggregated. Second, in any game where a player i could potentially solve the puzzle first (i.e., the maximum length of shortest paths of player i to other players is less than the maximum time T the game lasts), there is an SPNE where she does win, and further she wins in the shortest possible amount of time. Third, for a group of communication networks that includes graphs like the tree and the complete graph, then every SPNE is solved by some player is no more than N-1 periods. Fourth, for other simple graph structures, there are SPNEs for which an arbitrary amount of time passes before some player solves the game.

The intuition for all of these results boils down to the following. Every complete graph involves at least two agents connected to each other who will potentially each hold every piece of information the opponent lacks. When this happens, we are in the normal Game of Chicken. Since the problem has a final period T and we are looking for SPNE, in the final period T the two players just play chicken with each other, and chicken has two pure strategy Nash equilibria: I go straight, you swerve, or you go straight and I swerve. Either way, one of us “swerves”/shares information, and the other player solves the puzzle. The second theorem just relies on the strategy where whichever player we want to solve the puzzle refuses to share ever; every other player can only win nonzero payoff by getting their information to her, and they want to do so as quickly as possible. The fourth result is pretty interesting as well. Consider a 1000 period game, with four players arranged in a square: A talks to B and D, B talks to A and C, C to B and D, and D to A and C. We can be in a situation where B needs what A has, and A needs what B has, but not be in a duel. Why? Because A may be able to get the information from C, and B the information he needs from D. Consider the following hypothesized SPNE, though: everyone hides until period 999, then everyone passes information on in 999 and 1000. In this SPNE, everyone solves the puzzle simultaneously in period 1000 and gets one-fourth of the bonus reward. If any player deviates and, say, shares information before period 999, then the other players all play an easily constructed strategy whereby the three of them solve the following period but the deviator does not. If the reward is big enough, then all the discounting we need to get to period 1000 will not be enough to make anyone want to deviate.

What does this all mean for social science? Essentially, if I want information to be shared and I have both team and individual bonuses, then no matter what individual and team bonuses I give, the information will be properly aggregated by strategic agents quite quickly if I make communication follow something like a hierarchy. Every (subgame perfect) equilibrium involves quick coordination. On the other hand, if the individual and team bonuses are not properly calibrated and communication involves cycles, it may take arbitrarily long to coordinate. I think a lot more could be done with these ideas applied to traditional team theory/multitasking.

One caveat: I am not a fan at all of modeling this game as having a terminal period. The assumption that the game ends after T periods is clearly driving the result, and I have some hunch that simply using a different equilibrium concept than SPNE and allowing an infinite horizon, you could solve for very similar results. If so, that would be much more satisfying. I always find it strange when hold-up problems or bargaining problems are modeled as having necessarily a firm “explosion date”. This avoids much of the great complexity of negotiation problems!

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Matthew Elliott is another job market candidate making the rounds this year, and he presented this nice paper on matching today here at Northwestern. In the standard bilateral search model (due to Hosios), firms and workers choose whether or not to enter the job market (paying a cost), then meet sequentially with some probability and bargain over the wage. In these models, there can either be too much entry or too little; an additional unemployed worker entering makes it easier for firms to find an acceptable worker but harder for other unemployed workers to find a job. This famous model (itself an extension of Diamond-Mortensen-Pissarides, they of the 2010 Nobel) has been extended to allow for search costs, on-the-job search and trilateral bargaining, where two firms fight over one worker. Extending it to the most general case, where n firms and m workers, perhaps varying by worker and firm, constructed stochastically, is a problem which required much more advanced tools in network theory.

Elliott provides those results, remaining in the framework of negotiated rather than posted wages; as he notes, this theory is perhaps more applicable to high-skill labor markets where wages are not publicly posted, and workers are heterogeneous. Workers and firms simultaneously decide whether to enter the job market (paying a cost) and how hard to search (in an undirected manner). Workers match stochastically with each firm (who desires to hire one worker) depending on the level of search. Firms then negotiate how to split the surplus the match will generate, and some agent is hired.

If this sounds like Shapley-Shubik assignment to you, you’re on the right track. Because we’re in a Shapley-Shubik world, pairwise stability of the final assignment places us in the core; there are no deviations, even coalitional deviations, available. In a companion paper, Elliott shows that the assignment can be decomposed into the actual assigned links and the “best outside option” link for each agent. The minimum pairwise stable payoff can be found by adding and subtracting the values of each agent’s chain of outside option links.

The results for the labor market are these: there is never too much entry, search can sometimes be too heavy (though never too light), and that the labor market is “fragile”; it can unravel quickly. Entry is efficient because a new entrant will only change payoffs if he forces an old link to sever. By the definition of pairwise stability, the firm and the worker from that link must collectively be getting a higher payoff if they sever, since otherwise they would just reform their old link. That is, new entrants only thicken the market. Unlike in Hosios, since entering firms in Elliott have to bid up the wage of a worker they want in order to “steal” him from his current match, their effect on other firms when they enter does not cause a negative externality: they pay for causing the externality. The same argument in reverse applies to workers. Search is too heavy because having more outside options allows you to, in some sense, negotiate away more of the surplus from your current match. Labor market fragility occurs because, mathematically, everyone is getting payoffs that result from a weighted, connected graph. If one agent decides not to enter (his entry costs rise by epsilon), then the outside option of other agents is lowered. Their own current links are therefore willing to give them less of the surplus. Because of this, they may choose not to enter, and so on down the line. That is, minor shocks to the search process can create the necessary amplifications seen in the business cycle.

It would be nice to extend this type of model of the labor market to its dynamic setting – it’s not clear to me how sensible it is to talk about labor markets unraveling when all choices are being made simultaneously. Nonetheless, this paper does provide continuing metaproof about the usefulness of network theory and matching to a wide range of economic problems. The operations research types still no doubt have a lot to teach economists.

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Aumann (1976) famously proved that with common priors and common knowledge of a posterior, individuals cannot agree to disagree about a fact. Geanakoplos and Polemarchakis (1982) explained how one might reach a common posterior, by showing that if two agents can communicate their posterior, and then reupdate, they will in finite transfers of information converge on a posterior belief (of course, it might not be the true state of the world that they converge on, but converge they will), and hence will not agree to disagree. This fact turns out to generalize to signal functions other than Bayesian updates, as in Cave (1983).

One might wonder, then: does this result hold for more than two people? The answer is that it does not. Parikh and Krasucki define a communication protocol among N agents as a sequence of names r(t) and s(t) specifying who is speaking and who is listening in every period t. Note that all communication is pairwise. As long as communication is “fair” (more on this shortly), meaning that everyone communicates with everyone else either directly or indirectly an infinite number of times, and as long as the information update function satisfies a convexity property (Bayesian updating does), then beliefs will converge, although unlike in Cave, Aumann and Geanakoplos, the posterior may not be common knowledge.

There is no very simple example of beliefs not converging, but a long(ish) counterexample is found in the paper. A followup by Houy and Menager notes that even when information updates are Bayesian, different order of communication can lead to beliefs converging to different points, and proves results about how much information can be gleaned when we can first discuss in which order we wish to discuss our evidence; if it is common knowledge that two groups of agents disagree about which protocol will make them better off (in the sense of giving them the finest information partition after all updates have been done), then any order of communication, along with the knowledge about who “wanted to speak first”, will leads to beliefs converging to the same point. That is, if Jim and Joe both wish to speak second, and this is common knowledge, then no matter who speaks first, beliefs will converge to the same point).

One important point about Parikh and Krasucki. First, the result is “wrong” in the sense that the method of updating beliefs is problematic. In particular, when agents get new information, the method of updating beliefs turns out to ignore some valuable information. I will make this statement clearer in a post tomorrow.

This entire line of reasoning makes you wonder whether, under common topologies of communication, we can guarantee convergence of beliefs in groups: or indeed, whether we can guarantee that “the boss”, somehow defined, knows at least as much as everyone else. This is the project I’m working on at current, and hopefully will have results to share here by the end of the year.

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The basic economic model is often called “selfish”, but this is a misnomer. What separates economics from other cultural sciences is the assumption that humans are optimizers – that is, that they have an objective which they attempt to maximize. For this reason, economics has traditionally been wary of models, common in other social sciences, that apply genetic evolution theory to cultural objects. Traditionally, this has meant something along the lines of “if A interacts with B, A’s meme spreads to B with some probability” and then investigating the diffusion through society. With such a model, altruism could diffuse, for instance, if it helps the group as a whole survive. This is directly opposed to the idea that humans are maximizers, since regardless of group benefit, individuals may not wish to go along with “cooperate” just because they have heard the idea.

Nonetheless, this recent review by three Belgian social scientists argues that evolutionary biology has a lot to teach social science. In particular, they offer well-developed models of simulation. To the extent that economics, in the aggregate, *looks like* meme diffusion, then these models can still be useful. Therefore, I see proving equivalence between individual self-interested behavior and memetic spread as a useful direction for economists to investigate; for instance, is there a way to collapse, for some class of utility maximization, innovation choice such that at the population level innovation diffusion under utility maximization just looks like memetic diffusion?

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Monetary incentives, particularly as to how they relate to agency problems and free-rider problems, are relatively well understood compared to social incentives, such as the desire to socialize with friends in the workplace, or the desire to work hard when one witnesses ones peers working hard. Such data is hard to come by, however. In the new issue of ReStud, Bandiera et al gather data on friendship relations among immigrant workers at a fruit farm in the UK. Each day, workers are assigned to work in various parts of the field, sometimes with friends and sometimes not. Using variation in field quality on different days as an instrumental variable, the authors find that the average worker does not become more productive when working alongside friends, but this happens because high productivity workers work less hard when among friends, and low productivity workers work harder when among friends (on average, the effect is 10% either way). The change is productivity is monotonic in the difference in innate ability. Should the employer be able to optimally allocate workers alongside friends to maximize the effect of positive social incentives, productivity would increase 15%. As a comparison, earlier work has shown that switching from weak to strong monetary incentives (such as fixed wage to piece rate) has been shown to increase productivity by around 20%. That is, the effect of optimal social incentives is strong even relative to optimal monetary mechanism design.

This paper is very closely related to the 2009 AER “Peers at Work” by Mas & Monetti. That paper finds similar results – agents work harder when they witness high productivity workers.

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From a special issue on model building in the journal Social Networks, this paper by Robins et al discusses statistical modeling of networks as a tool both to identify whether structures in network data are due to chance or not, and as a way to model hypotheses about the global effects of local network structures. There is a good background on the ERG, or p* model, which allows a very flexible modeling of dependence relations among edges in the graph (for instance, balance theory). The essential result of ERG is the Hammersley-Clifford Theorem, which states that IFF a graph is Markovian, in that the probability of a tie between two nodes is dependent only on a local subset of the full graph, then the probability of the tie can be expressed as an exponential function of the hypothesized local dependencies.